I understand that the Hopf fibration $\pi: \mathbb{S}^3 \to \mathbb{S}^2$ is a kind of prequantum bundle. Is it possible to extend it to a map from $\mathbb{S}^3$ to any implicit surface $F(x, y, z)=0$?
To be more precisely, let $Q$ be a 3-dimensional manifold (for example $\mathbb{S}^3$) with $\alpha$ a fixed 1-form, $\Sigma$ be a 2-dimensional manifold given by $F(x, y, z)=0$ (for example a torus or a hyperboloid) with $\sigma$ a fixed 2-form, is there any map $\pi: Q \to \Sigma$ such that $\textrm{d}\alpha = \pi^*\sigma$ (here $^*$ is the pullback operator)?